Problem 1 - Advanced Engineering Mathematics

Question

State the size of the given matrix. $$ \left(\begin{array}{llll} 1 & 2 & 3 & 9 \\ 5 & 6 & 0 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is 2x4.

Step 1: Define the Problem

We need to determine the size of the given matrix. A matrix's size is defined by the number of its rows and columns.

Step 2: Count the Rows

Examine the given matrix:\[\begin{array}{llll}1 & 2 & 3 & 9 \5 & 6 & 0 & 1 \\end{array}\]The matrix has 2 horizontal arrangements of numbers, indicating there are 2 rows.

Step 3: Count the Columns

Count the vertical arrangements of numbers in the matrix:\[\begin{array}{llll}1 & 2 & 3 & 9 \5 & 6 & 0 & 1 \\end{array}\]Each row contains 4 numbers, indicating there are 4 columns.

Step 4: State the Size of the Matrix

Combine the number of rows and columns to state the size of the matrix: The matrix has 2 rows and 4 columns, so it is a 2x4 matrix.

Key Concepts

Understanding Matrix Notation

Matrix notation is a method of writing a set of numbers or expressions in a simple rectangular form. This helps us visualize and work with the data easily. A matrix is represented by a capital letter, such as \( A \) or \( B \). Inside the brackets, you see a grid-like arrangement:

• The individual elements (numbers) are typically represented by small letters with subscripts, such as \( a_{ij} \), where \( i \) denotes the row and \( j \) denotes the column.
• Each number in a matrix is called an element or an entry.
• The entire matrix can be enclosed with parentheses, brackets, or braces, but parentheses ( ) are more commonly used.

In this specific example matrix, the notation is:\[\left(\begin{array}{llll} 1 & 2 & 3 & 9 \ 5 & 6 & 0 & 1 \end{array}\right)\]The elements of the matrix are aligned in a rectangular shape using rows and columns, making it easier to specify and manage complex data sets.

Matrix Rows and Columns Explained

In mathematics, understanding rows and columns is crucial when dealing with matrices. Rows are horizontal lines in the grid, and columns are vertical lines. Here’s how to identify them:

• **Rows**: A row is a horizontal strip of data. In our example, this matrix has two sets of horizontal arrangements. The first row contains [1, 2, 3, 9], while the second row contains [5, 6, 0, 1]. Always start counting rows from the top.
• **Columns**: A column is a vertical line of data. To count columns, observe how many numbers are present in each row vertically. In this matrix, there are four columns. The first column contains numbers [1, 5], the second column [2, 6], and so on. Columns are counted from left to right.

Remember that visually distinguishing between rows and columns helps in determining operations such as addition or multiplication of matrices. It also aids in understanding matrix transformations and other applications.

How to Determine Matrix Dimensions

One of the basic tasks with matrices is determining their dimensions. Dimensions refer to the number of rows and columns in a matrix and are usually denoted as \( m \times n \), where \( m \) represents the number of rows and \( n \) the number of columns.

Steps to Determine Dimensions:

1. **Count the Rows**: Look horizontally in the matrix arrangement. Count how many such horizontal arrangements (rows) exist. In our given matrix, there are 2 rows.
2. **Count the Columns**: Focus vertically on the arrangement. Count the number of vertical data columns. In our example, each row has 4 numbers, indicating there are 4 columns.
3. **Combine the Counts**: Put the counts together in the format \( m \times n \) . Using the provided example, this matrix is a 2x4 matrix.

Knowing the dimensions of a matrix not only helps in identifying its size but is also essential when performing operations, as operations such as addition and multiplication are dependent on the matrices having compatible dimensions.

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